Analytical Study of Semi-Discrete Hamiltonian Dynamics in Diverse Potential Landscapes with Discrete Symmetries

I present a comprehensive analytical investigation of semi-discrete time-stepping methods applied to Hamiltonian systems in two spatial dimensions. This study explores the dynamics of a particle subjected to five distinct potential functions, encompassing both continuous potentials with one discretized dimension and fully discrete potentials defined on a two-dimensional lattice. The semi- discretization schemes combine continuous evolution for momenta (and one spatial coordinate in some cases) with probabilistic hopping rules for the discrete spatial variables. For each potential, I analyze the preservation of inherent discrete symmetries (including reflection, translation, and permutation), the behavior of conserved or invariant quantities, the characteristics of fixed points, and the properties of the probabilistic transitions governing the discrete motion. This comparative analysis across a range of potential landscapes provides valuable insights into the applicability and dynamical features of such hybrid discretization techniques for modeling physical systems with various forms of discrete symmetry.